Page 150 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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JWBK119-10
Illustration of the Two Methodologies Using a Case Study Data set 135
Summary for Transformed Data
Anderson-Darling Normality Test
A-Squared 0.29
P-Value 0.597
Mean 0.000528
StDev 0.000146
Variance 0.000000
Skewness −0.198567
Kurtosis −0.205168
N 30
Minimum 0.000231
1st Quartile 0.000451
Median 0.000538
3rd Quartile 0.000641
Maximum 0.000835
0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008
95% Confidence Interval for Mean
0.000474 0.000583
95% Confidence Interval for Median
0.000477 0.000591
95% Confidence Interval for StDev
95% Confidence Intervals 0.000117 0.000197
Mean
Median
0.00048 0.00050 0.00052 0.00054 0.00056 0.00058 0.00060
Figure 10.3 Histogram of case study data after Box--Cox transformation.

where N is the sample size. Then

⎧ 2 2 2 2
⎪ exp(1.2937 − 5.709A + 0.0186(A ) ), if 13 > A > 0.600
⎪
exp(0.9177 − 4.279A − 1.38(A ) ), if 0.600 > A > 0.340
⎨ 2 2 2 2
p = 2 2 2 2
⎪ 1 − exp(−8.318 + 42.796A − 59.938(A ) ), if 0.340 > A > 0.200
⎪
1 − exp(−13.436 + 101.14A − 223.73(A ) ), if A < 0.200
⎩ 2 2 2 2
Generally if the p-value of the test is greater than 0.05, we do not have enough evidence
to reject the null hypothesis (that the data is normally distributed).
Any statistical software can again easily do the above. Figure 10.3 was obtained by
MINITAB 14, which provides both the histogram plot with a normal curve on top of
it and the Anderson--Darling normality test on the same page. From the Anderson--
Darling test we can conclude that the transformed data is normally distributed as the
p-value was 0.597, way above the critical value of 0.05.

10.2.1.3 Estimate the process capability using the transformed data
Having successfully transformed the data, we can estimate the process capability us-
ing the formula below with the transformed data and the transformed specification,
91 −3 = 1.327 × 10 −6 (note that as the power is negative, the original upper specifica-
tion limit becomes the lower specification limit):

¯ x − LSL
C pk = .
3σ
For our example data set C pk = 1.19 (see Figure 10.4).